Categorical Extension of Dualities: From Stone to de Vries and Beyond, I

TL;DR

This paper develops a general categorical framework to extend dualities in topology, providing new proofs and duality theorems for categories of compact Hausdorff and Tychonoff spaces, enhancing the understanding of their dualities.

Contribution

It introduces a new categorical approach to extend dualities, offering alternative proofs and new duality theorems for $f KHaus$ and $f Tych$ categories.

Findings

New proof of de Vries Duality Theorem for $f KHaus$

Alternative proof of duality extension to $f Tych$

Derived new duality theorems for both categories

Abstract

Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category $\bf KHaus$ of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category $\bf Tych$ of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, $\bf{KHaus}$ and $\bf Tych$.